This follows again from the general theory of curves of genus 1, the proof by PonceletPoncelet considered actually a more general dynamics involving arbitrary pairs of non-tangent conics however predating such facts.
See [1]}, p. 58 (especially Cor. 4.5), for a somewhat different description of the translation, without invoking elliptic curves, and described as an element of \(S_1\) . This is consistent because the connected component of the identity in the group of real points of an elliptic curve over \({R}\)  is a circle. The present description covers also the case of complex coordinates. (Compare with Remark REF  above.)
